Milnor's isospectral tori and harmonic maps

Abstract: A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold $(M,g)$ determines the Riemannian metric $g$ up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given Riemannian manifold $(\Sigma,h)$ to $M$ determines the Riemannian metric on the target space. We consider this question in the case of harmonic maps between flat tori. In particular, we show that the two isospectral, non-isometric $16$-dimensional flat tori found by Milnor cannot be distinguished by the energy spectrum of harmonic maps from $d$-dimensional flat tori for $d\leq 3$, but can be distinguished by certain flat tori for $d\geq 4$. This is related to a property of the Siegel theta series in degree $d$ associated to the $16$-dimensional lattices in Milnor's example.